Download Fixed Point in Minimal Spaces

Nonlinear Analysis: Modelling and Control, 2005, Vol. 10, No. 4, 305–314
Fixed Point in Minimal Spaces
M. Alimohammady1 , M. Roohi2
1
Department of Mathematics, Mazandaran University, Babolsar, Iran
[email protected]
2
Islamic Azad University-Sari Branch, Iran
[email protected]
Received: 24.08.2005
Accepted: 10.10.2005
Abstract. This paper deals with fixed point theory and fixed point property
in minimal spaces. We will prove that under some conditions f : (X, M) →
(X, M) has a fixed point if and only if for each m-open cover {Bα } for X
there is at least one x ∈ X such that both x and f (x) belong to a common Bα .
Further, it is shown that if (X, M) has the fixed point property, then its minimal
retract subset enjoys this property.
Keywords: fixed point, orbits, multifunction, minimal space.
1 Introduction and preliminaries
Fixed point theory is a very attractive subject, which has recently drawn much
attention from the communities of physics, engineering, mathematics etc. In this
field, there have been many representative approaches method by orbits [1]. In
[2], authors used fixed point theory to find a method to estimate the optimum
neighborhood with the chosen gain matrix.
In this paper, we prove some results too stunning not be in the spotlight.
These results are typical of the most attractive aspects of the fixed point theory
in minimal spaces in that they are proved. We show any retract subset of a space
with fixed point property would have the fixed point property.
In 1950, H. Maki, J. Umehara and T. Noiri [3] introduced the notions of
minimal structure and minimal space. They achieved many important results
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compatible by the general topology case. Some other results about minimal spaces
can be found in [4–9].
For easy understanding of the material incorporated in this paper we recall
some basic definitions. For details on the following notions we refer to [4], [3]
and [7].
Definition 1. [3] A family M ⊆ P(X) is said to be minimal structure on X if
∅, X ∈ M. In this case (X, M) is called a minimal space. Throughout this paper
(X,M ) or (Y, N ) means minimal space.
Example 1. [3] Let (X, τ ) be a topological space. Then M = τ, SO(X),
P O(X), αO(X) and βO(X) are examples of minimal structures on X.
Definition 2. [3] A set A ∈ P(X) is said to be an m-open set if A ∈ M.
B ∈ P(X) is an m-closed set if B c ∈ M. We set
[
m − Int(A) = {U : U ⊆ A, U ∈ M},
\
m − Cl(A) = {F : A ⊆ F, F c ∈ M}.
Remark 1. Choosing one of the τ, SO(X), P O(X), αO(X) and βO(X) instead
of M, then m − Int(A) would be Int(A), sInt(A), pInt(A), αInt(A) and
βInt(A) respectively. Similarly, m − Cl(A) is equal to Cl(A), sCl(A), pCl(A),
αCl(A) and βCl(A) respectively.
Proposition 1. [3] For any two sets A and B,
(i) m − Int(A) ⊆ A and m − Int(A) = A if A is an m-open set;
(ii) A ⊆ m − Cl(A) and A = m − Cl(A) if A is an m-closed set;
(iii) m − Int(A) ⊆ m − Int(B) and m − Cl(A) ⊆ m − Cl(B) if A ⊆ B ;
(iv) m − Int(A ∩ B) = (m − Int(A)) ∩ (m − Int(B)) and (m − Int(A)) ∪
(m − Int(B)) ⊆ m − Int(A ∪ B);
(v) m − Cl(A ∪ B) = (m − Cl(A)) ∪ (m − Cl(B)) and m − Cl(A ∩ B) ⊆
(m − Cl(A)) ∩ (m − Cl(B));
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(vi) m − Int(m − Int(A)) = m − Int(A) and m − Cl(m − Cl(B)) =
m − Cl(B);
(vii) (m − Cl(A))c = m − Int(Ac ) and (m − Int(A))c = m − Cl(Ac ).
Definition 3. [7] A minimal space (X, M) enjoys the property U if the arbitrary
union of m-open sets is an m-open set.
Proposition 2. [7] For a minimal structure M on a set X, the following are
equivalent.
(i) M has the property U .
(ii) If m − Int(A) = A , then A ∈ M.
(iii) If m − Cl(B) = B , then B c ∈ M.
Definition 4. Let (X, M) and (Y, N ) be two minimal spaces. We say that a
function f : (X, M) → (Y, N ) is a minimal continuous (briefly m-continuous) if
f −1 (B) ∈ M, for any B ∈ N .
The following results are the immediate consequences of Definition 4.
Proposition 3. Suppose (X, M) and (Y, N ) are minimal spaces. Then
(i) the identity map idX : (X, M) → (X, M) is m-continuous;
(ii) idX : (X, M) → (X, N ) is m-continuous where (X, M) and (X, N ) are
minimal spaces and N ≤ M;
(iii) any constant function f : (X, M) → (Y, N ) is m-continuous.
Theorem 1. The composition of two m-continuous functions is an m-continuous
function.
2 Orbits and fixed point
For two sets X and Y and each element x of X we associate a nonempty subset
F (x) of Y and this correspondence x 7→ F (x) is called a multi-valued mapping
or a multifunction from X into Y ; i.e., F is a function from X to 2Y \ {∅} and is
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denoted by F : X → 2Y . The lower inverse of a multi-valued mapping F is the
multi-valued mapping F l of Y into X defined by
F l (y) = x ∈ X : y ∈ F (x) ,
also for any nonempty subset B of Y we have,
F l (B) = x ∈ X : F (x) ∩ B 6= ∅ ,
finally it is understood that F l (∅) = ∅. The set {x ∈ X : F (x) ⊆ B} is the
upper inverse of B and is denoted by F u (B). f is minimal lower semicontinuous
(m.l.s.c.), if for every U ⊆ Y m-open, f l (U ) is m-open in X.
Let f : X → 2Y and g : Y → 2Z be two multifunctions. The composition
gof : X → 2Z is defined by
gof (x) =
[
g(y).
y∈f (x)
Definition 5. [7] For a minimal space (X, M),
(i) a family of m-open sets A = {Aj : j ∈ J} in X is called an m-open cover
S
of K if K ⊆ j Aj . Any subfamily of A which is also m-open cover of K
is called a subcover of A for K;
(ii) a subset K of X is m-compact whenever given any m-open cover of K has
a finite subcover;
(iii) (X, M) is said to be m − T2 space if for each distinct points x, y ∈ X, there
exists U, V ∈ M containing x and y respectively, such that U ∩ V = ∅.
In the following lemma we show the equivalence of point-wise m-continuity
and m-continuity which has a key role for our result.
Lemma 1. Suppose f : X → Y is a function, where (X, M) and (Y, N ) are two
minimal spaces. Then f is point-wise m-continuous if f is m-continuous.
Proof. Assume f is point-wise m-continuous, x ∈ X, V ∈ N and f (x) ∈ V .
Then x ∈ W = f −1 (V ) ∈ M. Therefore, f (W ) ⊆ V .
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Definition 6. A minimal space (X, M) enjoys the property I if the finite intersection of m-open sets is an m-open set.
Theorem 2. Suppose M is a minimal structure with property I on X, (X, M) is
m − T2 space and f : (X, M) → (X, M) is m-continuous. Then f has a fixed
point if and only if for each m-open cover {Bα : α ∈ A} of X there is x ∈ X and
α ∈ A such that both x and f (x) lie in Bα .
Proof. One direction is straightforward. For the converse, suppose that f has no
fixed point. Then x 6= f (x) for each x ∈ X. Since f is m-continuous, X is m−T2
and M has property I, so there is Wx and Ux in M containing respectively x and
f (x) such that Ux ∩Wx 6= ∅ and f (Wx ) ⊆ Ux . Now {Wx : x ∈ X} is an m-cover
of X, so there is z ∈ X and x0 ∈ X such that Wx0 contains both z and f (z).
Since z ∈ Wx0 , so f (z) ∈ f (Wx0 ) ⊆ Ux0 . On the other hand f (z) ∈ Wx0 , so
f (z) ∈ Wx0 ∩ Ux0 which is impossible.
Extending one direction of Theorem 2 is our next task.
Theorem 3. Suppose F : X → 2X is a multifunction such that:
(i) for each m-open cover {Wα : α ∈ A} for X there are z ∈ X and α0 ∈ A
for which Wα0 contain both z and F (z) (i.e., z ∈ Wα0 and F (z) ⊆ Wα0 );
(ii) if z ∈
/ F (z) then there are Wz and Uz of M with {z} ⊆ Wz and f (z) ⊆ Uz
such that Uz ∩ Wz = ∅.
Then F has a fixed point.
Proof. On the contrary, if z ∈
/ F (z) for each z ∈ X then from (ii) there are
Wz and Uz with mentioned properties. Then {Wz : z ∈ X} is an m-open cover
for X so from (i) there are two elements x0 , z0 ∈ X such that both {x0 } and
F (x0 ) are contained in Wz0 . On the other hand, x0 ∈ Wz0 implies that F (x0 ) ⊆
f (Wz0 ) ⊆ Uz0 . Since F (x0 ) ⊆ Wz0 , so F (x0 ) ⊆ Uz0 ∩ Wz0 = ∅ which is a
contradiction.
Definition 7. A family {Aj : j ∈ J} in P(X) has the finite intersection property
if any its finite subfamily has nonempty intersection.
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Our next result indicates the relation of fixed point and orbits of a multifunction.
Proposition 4. Suppose F : X → 2X is a multi-valued map and there is
x0 ∈ X such that O(x0 ) has finite intersection property. Then F has a fixed
point if O(F 2 (x)) ⊆ F (x) for all x ∈ X.
Proof. It is easy to see that F (O(x0 )) ⊆ O(x0 ), so
K = A ⊆ O(x0 ) : A 6= ∅, F (A) ⊆ A
is a nonempty set. Partially ordered K by inclusion. Since O(x0 ) has finite
intersection property, so from Zorn’s lemma K has minimal element, say C.
F (C) ⊆ C and F (F (C)) ⊆ F (C) imply that F (C) = C. Now, if u ∈
/ F (u)
for each u ∈ C, then u ∈
/ O(F 2 (u)). F (u) ⊆ F (C) = C follows from the fact
that u ∈ C, therefore F k (u) ⊆ C for any nonnegative integer k. O(F 2 (u)) = C
can be derived from minimality of C. Consequently, u ∈ O(F 2 (u)) which is a
contradiction.
We are ready to extend a result due to Ciric [10].
Definition 8. Suppose (X, M) is a minimal space. A subset A of X is said to be
have minimal closure finite intersection property if the intersection of elements of
any family A = {m − Cl(Aα ) ⊆ A : α ∈ I} is nonempty, where any its finite
intersection of elements of A is nonempty.
Theorem 4. Suppose (X, M) is a minimal space, F : X → 2X is a multifunction
and
(i) there is x0 ∈ X such that m − Cl(O(x0 )) has minimal closure finite
intersection property,
(ii) m − Cl(O(F 2 (x))) ⊆ F (x) for all x ∈ X,
(iii) F (m − Cl(O(x0 ))) ⊆ m − Cl(O(x0 )).
Then F has fixed point.
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Proof. Set K = {m − Cl(A) : A ⊆ O(x0 ), F (m − Cl(A)) ⊆ m − Cl(A)}
which is a nonempty set by (iii). K is a partially ordered set by inclusion. Then
K has a maximal element by Zorn’s lemma. We denote this maximal element
by m − Cl(B). Consequently, F (m − Cl(B)) ⊆ m − Cl(B) and so F (m −
Cl(B)) = m − Cl(B). Now if x ∈
/ F (x) for all x ∈ B, then (ii) implies that x ∈
/
2
2
2
m − Cl(O(F (x))). But x ∈ B, so O(F (x)) ⊆ B, thus m − Cl(O(F (x))) ⊆
m − Cl(B). Then
F m − Cl O F 2 (x)
⊆ F 2 (x) ⊆ m − Cl O F 2 (x) .
Therefore, O(F 2 (x)) = B concludes from maximality of B, so x ∈ O(F 2 (x))
which is a contradiction.
An immediate consequence of Theorem 4 can be state in the following.
Corollary 1. Suppose (X, τ ) is a topological space, F : X → 2X is a set valued
map and
(i) there is x0 ∈ X such that O(x0 ) is compact,
(ii) O(F 2 (x)) ⊆ F (x) for all x ∈ X,
(iii) F (O(x0 ))) ⊆ O(x0 ).
Then F has fixed point.
Proof. It should be noticed that in topological space minimal closure finite intersection property is equivalent to the compactness. Applying Theorem 4 completes
the proof.
Definition 9. A function f : X → X is called strongly non-periodic if for every
x ∈ X, x 6= f (x) implies x ∈
/ Ō(f 2 (x)). A function f is said to be orbitally
continuous if for each x, y ∈ X, y = limi f ni (x) implies f (y) = limi f ni +1 (x).
Corollary 2. [10] Let X be a topological space and f : X → X be a strongly
non-periodic and orbitally continuous mapping. If for some x0 ∈ X the set Ō(x0 )
is compact, then there exist a cluster point x ∈ O(x0 ) such that f (x) = x.
Furthermore, if for every (x, y) ∈ X × X, x 6= y implies (f x, f y) 6= (x, y), then
x is a unique fixed point of f in X.
Proof. Apply Corollary 1 but for single valued map f .
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3 Fixed point property
Definition 10. (X, M) and (Y, N ) are called m-homeomorphic if there exists a
bijective function f : X → Y for which f and f −1 are m-continuous. In this case,
f is called an m-homeomorphism and X and Y are said to be m-homeomorphic.
Definition 11. (X, M) is said to have the fixed point property if every m-continuous function f : X → X has a fixed point.
Example 2. Suppose X = {x1 , x2 , x3 } and M = {∅, {x1 }, {x2 }, X} is a
minimal structure on X. In order to show that X has the fixed point property
it is enough to show that any function f : (X, M) → (X, M) which has not
fixed point is not m-continuous. If f has not fixed point, then f (x3 ) 6= x3
and then f (x3 ) = x1 or f (x3 ) = x2 . Therefore, x3 ∈ f −1 ({x1 }) and since
f −1 ({x1 }) ∈
/ M so f −1 ({x1 }) does not lie in M or x3 ∈ f −1 ({x2 }) ∈
/ M
which implies that f : (X, M) → (X, M) is not m-continuous.
Next result shows that fixed point property is invariant under m-homeomorphisms.
Proposition 5. Suppose X is m-homeomorphic to Y . Then Y has the fixed point
property if X has this property too.
Proof. Suppose h : X → Y is an m-homeomorphism and g : Y → Y is an mcontinuous function. Since h−1 ogoh : X → X is m-continuous, applying Theorem 1 and fixed point property of X, there exists x0 ∈ X in which h−1 ogoh(x0 ) =
x0 . Set y0 = h(x0 ), then h−1 og(y0 ) = x0 . Therefore, g(y0 ) = h(x0 ) = y0 which
is required.
Definition 12. A subset A of a minimal space (X, M) is a minimal retract of X
if there is an m-continuous function r : X → A by r(a) = a for all a ∈ A. In
this case, r is called minimal retraction.
In following we prove the fixed point property for some subset of a set with
this property.
Proposition 6. Suppose (X, M) has the fixed point property and A is a minimal
retract of X. Then A has the fixed point property too.
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Proof. Suppose f : A → A is an m-continuous function and r : X → A is a
minimal retraction. Consider the following compositions,
r
f
i
X → A → A → X,
where i is the inclusion map. From Proposition 3, i is an m-continuous function. Since X has the fixed point property, there exists x0 ∈ X such that
iof or(x0 ) = x0 and so f or(x0 ) = x0 . Put a = r(x0 ) ∈ A, then f (a) = x0 .
Consequently, r(f (a)) = r(x0 ) which implies that f (a) = a.
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